1. Can we probe quantum structure of space-time by black holes?
2008 YongPyong
Can we probe quantum structure of space-time by black holes?
Satoshi Iso
(KEK)

2. Plan of the talk [1] Overview of BH thermodynamics
　　 　・causal structure of horizon
　　　・Hawking radiation
　　　・stringy picture of BH entropy
[2]　Hawking radiation via gravitational anomalies
　　　・universality of Hawking radiation　　
[3] Conclusion
・ towards quantum nature of space-time

3. [1] Overview of black hole thermodynamics
Black hole was theoretically proposed as a strange
solution to the Einstein equation in 1916 by Schwarzschild.

4. Now it is known that our universe is full of black holes
from small size (star) to huge size (center of galaxy).
・Cyg X ~ Mo (=solar mass)
More than hundreds of compact BH will exist
in our galaxy.
・middle mass BH ~ 1000 Mo
exists in active galaxy(M82), but not in our galaxy
・Huge mass BH ~ Mo
AGN (center of galaxies)
They are very active (radiates X-rays) and
astro-physically very interesting objects.

5. But even from purely theoretical point of view,
black holes are very attractive objects
because it is believed that
we can probe the quantum structure of space-time
by investigating thermodynamic properties of BH.
Analogy with the discovery of quantum mechanics
Planck asked why the field theories
with infinite d.o.f. have finite specific heat?
---> energy quanta
Now we can ask why BH has a finite entropy
proportional to the area?
--> space-time quanta?

6. Schwarzschild black holes
horizon radius:
Curvature:
(1) Curvature singularity at r=0.
(2) Horizon of the space-time at r=rH.

7. Causal structure of horizon*
Tortoise coordinate: r
horizon
Null coordinates:

8. Kruskal coordinates: (surface gravity= ) Metric is now regular
in the Kruskal coord. V t U
r=0
U=0, V=0 at horizon
II: BH
U=0 future horizon
V=0 past horizon
III
I: exterior region
IV: WH
r=const
r=0

9. Horizon is a null hypersurface.
Even light cannot come out of the horizon.
rH =2GM
BH mass always increases classically.
Horizon area　never decreases
like entropy in thermodynamics.
d A > 0 =

10. Analogy with Thermodynamics
Equilibrium
Thermodynamics
Black Hole
0th law
T=const.
κ=const.
1st law
dU = TdS + ・・
dM =κ/(8πG) dA+ ・・
2nd law
dS > 0
dA > 0 = =
Classical correspondence

11. Hawking radiation from black hole
In 1974 Hawking found that black hole radiates.
This really gave the meaning to the thermodynamics of BH.
Hawking temperature:
Entropy of BH:
They are quantum effects!

12. (comparable to entropy of universe)
For BH with 10 solar mass -9
TH ~ 6×10 K
SBH ~ kB
very low temperature 79
huge entropy
(comparable to entropy of universe) 58
cf. Entropy of sun ~ 10

13. In the classical limit, TH
SBH ∞
Hawking radiation = one-loop quantum effect of matters
Entropy puzzle
　BH entropy is neither 0 nor infinity.
(1) BH entropy seems to contradict with the fact that
BH is classically unique due to the no-hair theorem.
(2) Also it is puzzling why it is proportional to area instead of
the volume of the BH.  holography

14. virtual pair creation of particlesBH -E E
real pair creation
Hawking radiation
(thermal)
Blackholes eventually evaporate.
information paradox
(violation of unitary evolution, i.e. pure state to mixed one)

15. Information paradox
If semiclassical description is valid near the horizon,
an initial pure state (collapsing matters) form a BH and
then it evaporates thermally to a mixed state of radiation.
 violation of quantum theory ??
Hints to a resolution
(1) black hole has a finite entropy
Poincare recursion time must be finite.
(like a decay of massive particle in a large but closed system)
(2) AdS /CFT: gauge theory is believed to describe gravity.
Gauge theory description of BH formation and evaporation?

16. How can we make BH from strings? statistical mechanics of BH
Strings: both of matters and space-time (graviton) are
excitations of strings
(4d) Newton constant G ~ (gs ls ) 2
rH =2GM
string
At strong coupling, string with mass M
becomes BH when its Schwarzschild radius
equals to the string length.
(2GM ~ ls)
S = kB log N(M) =kB ls M/ h
N(M) = exp (ls M/ h) 2
~ kB (GM) / (h G) =SBH

17. Extrapolation to strong coupling is not reliable.
Instead of fundamental strings, we can use
specific D-brane configurations.
(D1+D5+momentum along D1)
In this way, BH entropy can be understood
microscopically in string theory. (Strominger-Vafa)
Furthermore Hawking radiation can be also
understood as a unitary process of
closed string emission from D-branes.

18. Is everything understood in strings?
There is a missing link between BH and D-branes.
Once D-branes are in the horizon, they are invisible
from outside the BH.
Why are these d.o.f seen as entropy to an outside observer?
Information paradox is not yet well understood.
An interesting idea is a fuzzball proposal by Mathur,
which may interpolate macroscopic picture of black holes
and microscopic picture by D-branes.
Space-time is conjectured to by obtained by
coarse-graining the fuzzballs.

19. Horizon is responsible for thermodynamic properties of BH.
Horizon is not specific to black holes.
・ Rindler horizon
An accelerating observer can detect thermal
distribution proportional to his/her acceleration.
・ Cosmological horizon (de Sitter)
De Sitter space has a cosmological horizon,
and dS space has temperature and entropy.
Black holes are playgrounds for exploring
the quantum structure of space-time.

20. Horizon ~ dislocation of space-time ~ stratum
We may be able to probe the space-time itself
by looking at the dislocation (~horizon) of space-time.

21. In the following section,
we discuss the universality of the thermodynamic
quantities from the macroscopic point of view.
In particular, we will show that the Hawking radiation
can be universally understood
by the gravitational anomaly at the horizon.

22. [2] Hawking radiation and quantum anomalies
Robinson Wilczek (05)
Iso Umetsu Wilczek (06) BH
Quantum fields
in black holes.
Near horizon, each partial wave of d-dim quantum field
behaves as d=2 massless free field.
Outgoing modes = right moving
Ingoing modes = left moving
Effectively 2-dim conformal fields

23. Charged black hole (Ressner-Nordstrom solution).
Metric and gauge potential
r+: outer horizon
r-: inner horizon

24. -
Near horizon, potential terms can be suppressed.
Each partial wave behaves as d=2 conformal field.

25. once they are inside the horizon.
(2) Ingoing modes are decoupled
once they are inside the horizon.
These modes are classically irrelevant for the
physics in exterior region.
So we first neglect ingoing modes near the horizon.
The effective theory becomes chiral
in the two-dimensional sense.
gauge and gravitational anomalies
= breakdown of gauge and general coordinate invariance

26. (3) But the underlying theory is NOT anomalous.
Anomalies must be cancelled by quantum effects
of the classically irrelevant ingoing modes.
　　(～Wess-Zumino term)
flux of Hawking radiation

27. covariant gauge anomaly
U(1) charge current
total derivative
For outgoing modes
covariant gauge anomaly
flux at infinity
regularity at horizon
Flux can be given only by the information at the horizon
because anomaly is a total derivative.
S.I. Umetsu Wilczek(06)
simplifed by Banerjee Kulkarni(07)

28. Energy flow at infinity
total derivative
covariant gravitational anomaly
Energy flux can be given in terms of the surface gravity
at horizon!
Universality is assured by the anomaly structure.

29. Anomaly is a total derivative.
An integral of the instanton configuration gives the
topological number as the number of fermionic 0-mode.
Hawking radiation is determined by the information
at the horizon. (universality)
i.e. The flux is independent of the details of the metric
away from the horizon.

30. Higher-spin fluxes? yes Energy flux
Iso Morita Umetsu (07)
Higher-spin fluxes?
Energy flux
This is only a partial information of the Hawking radiation.
Can we generalize them to higher-spins and reproduce
thermal distribution ?
yes

31. Assuming that the effective field theory approximation
is good near the horizon,
thermal radiation is completely reproduced.
(This is a universal result as the anomalies are universal.)
How can we solve the information paradox?
The derivation of CFT near the horizon is based on
an assumption that there are no higher-derivative
terms in the action of the matter field.
If they exist, they become more and more important
near the horizon.
Loss of locality near the horizon.

32. [3] Summary
Most properties of BH are well-understood now
both macroscopically and microscopically…..
but
Black hole entropy
Hawking radiation
macroscopic
microscopically
universality
based on GR and QM
string theory
D-branes
missing link
entropy puzzle
information paradox

33. Since thermodynamic properties of BH are originated
in the horizon and horizons are general phenomena
in general relativity,
the missing link will be related to the quantum structure
of the space-time itself.
As blackbody radiation played an important role
in discovering the quantum mechanics,
black hole physics will play a similar role
to understand the quantum structure of space-time.
Still there are many mysteries.

34. Thank you for your attention.

35. ingoing
outgoing
For calculational convenience,
we divide the exterior region
into H and O. ＢＨ H
H: [r+, r+ + ε]
O: [r+ + ε , ∞] ε O
First neglect the classically irrelevant ingoing modes
in region H.

36. Gauge current and gauge anomaly
The theory becomes chiral in H. H O
Gauge current has anomaly in region H.
outer
horizon ε
consistent current
We can define a covariant current by
which satisfies

37. In region O,
In near horizon region H,
consistent current
= current at infinity
= value of consistent
current at horizon
are integration constants.
Current is written as a sum of two regions.
where

38. Variation of the effective action under gauge tr.
Using anomaly eq.
Impose　 δW　＋　δW’＝０
　　W’ = contribution from ingoing modes (WZ term)
cancelled by WZ term

39. ・Determination of
We assume that the covariant current should vanish at horizon.
Unruh vac.
Reproduces the correct Hawking flux

40. Total current including ingoing modes near the horizon
should be conserved!
ingoing mode
outgoing mode

41. EM tensor and Gravitational anomaly
Effective d=2 theory contains background of
graviton, gauge potential and dilaton.
Under diffeo. they transform
Ward id. for the partition function
=anomaly

42. Gravitational anomaly
consistent current
covariant current
In the presence of gauge and gravitational anomaly, Ward id. becomes
non-universal

43. Solve component of Ward.id.
(1) In region O
(2) In region H
(near horizon)
Using

44. Variation of effective action under diffeo.
(1)
(2)
(3)
(1) classical effect of background electric field
(2) cancelled by induced WZ term of ingoing modes
(3) Coefficient must vanish.

45. Reproduces the flux of Hawking radiation
Determination of
We assume that the covariant current to vanish at horizon.
since
we can determine
and therefore flux at infinity is given by
Reproduces the flux of Hawking radiation